Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Condensed matter

Spontaneous dimerization in the ground state of one-dimensional frustrated Heisenberg antiferromagnets

Alberto Pimpinelli

Published Feb 1, 1991• 6 citations
Publisher

Abstract

Starting from the exact result of Majumdar and Ghosh for the dimerized ground state of a Heisenberg linear-chain antiferromagnet with competing next-nearest-neighbor interaction ${\mathit{J}}_{2}$=1/2${\mathit{J}}_{1}$, it is shown that this state is an exact eigenstate of the same Hamiltonian with couplings extended up to fourth-nearest neighbors, if the relations ${\mathit{J}}_{1}$-2${\mathit{J}}_{2}$+${\mathit{J}}_{3}$=0 and ${\mathit{J}}_{3}$=2${\mathit{J}}_{4}$ hold. Furthermore, it is proved to all orders in perturbation theory that this state is the ground state of this Hamiltonian in a finite interval of values of the parameter \ensuremath{\lambda}==1/2-${\mathit{j}}_{2}$ around \ensuremath{\lambda}=0. The actual extension of this interval is computed numerically for finite chains and is shown to be significant.

DOI
10.1103/physrevb.43.3710

Related papers

Rigorous determination of spontaneously dimerized ground states in 1D Heisenberg antiferromagnets

Alberto Pimpinelli

Starting from the exact result of Majumdar and Ghosh (1969) for the dimerized ground state of a Heisenberg S=1/2 antiferromagnet with competing next-nearest-neighbour interaction J2+1/2J1 on a linear chain, it is shown that this state is the ground state of the same kind of Hamiltonian, if couplings are extended up to 2n-nearest-neighbours, and the relation J1=2n, J2=2n-1, ..., J2n=1 holds. Furthermore, it is proved rigorously that this state is the ground state when n=2 and J1-2J2+J3=0, J3=2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0<or=J4<or=1/4J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising model is discussed.

Journal of Physics Condensed Matter•Jan 28, 1991•5 citations

Spontaneously dimerized ground states in 1D frustrated Heisenberg antiferromagnets

Alberto Pimpinelli

We give an explicit proof that the completely dimerized (’Kekulè or nearest-neighbor resonant valence bond) state is the degenerate ground state of a linear chain S =1/2 Heisenberg antiferromagnet, with competing couplings extending up to 2n nearest neighbors, if the relation J1 = 2n, J2 = 2n − 1,...,J2n = 1, holds between the various exchange integrals. The previously known result for J2 =1/2 J1 is then just a special case corresponding to n=1. We study in detail n=2 (fourth-nearest neighbors) and show that this state is the ground state of the Hamiltonian when J1 − 2J2 + J3 = 0, J3 = 2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0 ≤ J4 ≤1/4 J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising models is made.

Journal of Applied Physics•Apr 15, 1991•2 citations

Frustrated spin-1/2 model in two dimensions with columnar dimer states as possible ground states

Uma Bhaumik, Indrani Bose

An antiferromagnetic spin S=1/2 model, the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$-${\mathit{J}}_{3}$-${\mathit{J}}_{4}$-${\mathit{J}}_{5}$ model, is considered in two dimensions. ${\mathit{J}}_{1}$,${\mathit{J}}_{2}$,${\mathit{J}}_{3}$,${\mathit{J}}_{4}$,${\mathit{J}}_{5}$ are the strengths of the nearest-neighbor, diagonal, next-nearest-neighbor, knight's-move-distance-away, and further-neighbor-diagonal interactions, respectively. It is known that when ${\mathit{J}}_{1}$:${\mathit{J}}_{2}$:${\mathit{J}}_{3}$:${\mathit{J}}_{4}$:${\mathit{J}}_{5}$=1:1:1/2:1/2:1/4 the four columnar dimer states are the exact eigenstates. In a bosonic mean-field theory, we find that the columnar dimer state is the ground state. We compare the results obtained with those obtained in the same mean-field theory for the one-dimensional Majumdar-Ghosh (MG) model. For special values of the couplings, the dimer state is known to be the exact ground state of the MG model. The comparison of the mean-field results for both the models leads to the conjecture that the columnar dimer states are the exact ground states of the ${\mathit{J}}_{1}$-${\mathit{J}}_{2}$-${\mathit{J}}_{3}$-${\mathit{J}}_{4}$-${\mathit{J}}_{5}$ model.

Physical review. B, Condensed matter•Nov 1, 1995•6 citations

A dimerized and frustrated spin model with alternating ferromagnetic and antiferromagnetic exchanges: investigating quantum phases of Heisenberg spin chains with s = 1/2 and 1

Shaon Sahoo, V. M. L. Durga Prasad Goli, Diptiman Sen +1 more

We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic ($J_1^F$) and antiferromagnetic ($J_1^A$) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction ($J_2^F$). In this model frustration is induced by the non-zero $J_2^F$ and the effect of dimerization is realized by changing the strength of $J_1^A$ for a fixed value of $J_1^F$. The model with site spin $s$ behaves like a Haldane spin chain with site spin 2$s$ in the limit of vanishing $J_2^F$ and large $J_1^F/J_1^A$. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of $J_1^A-J_2^F$ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.

•May 29, 2013•0 citations

Dimerization and effective decoupling in two spin-1 generalizations of the spin- 12 Majumdar-Ghosh chain

Natalia Chepiga, Frédéric Mila

We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, where ${J}_{2}$ is a next-nearest-neighbor Heisenberg coupling, and the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, where ${J}_{3}$ refers to a three-site interaction defined by ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin 1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model undergoes an effective decoupling into two next-nearest-neighbor (NNN) Haldane chains upon increasing ${J}_{2}$, while the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing ${J}_{3}$. By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, the former one adjacent to the Haldane one, the latter to the ferromagneric one, with an Ising transition between them. By contrast, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive ${J}_{2}$. In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.

arXiv:1907.11472•Physical review. B./Physical review. B•Sep 20, 2019•8 citations

Spontaneous dimerization in the S = 1 2 Heisenberg antiferromagnetic chain with competing interactions

F. D. M. Haldane

Spontaneous dimerization is found in the $S=\frac{1}{2}$ isotropic Heisenberg antiferromagnetic chain with competing nearest- and next-nearest-neighbor exchange, $\frac{{J}_{2}}{{J}_{1}}\ensuremath{\gtrsim}\frac{1}{6}$, and results from the same umklapp processes that lead to the N\'eel state when easy-axis exchange anisotropy is present. Spontaneous and externally induced dimerizations are contrasted.

Physical review. B, Condensed matter•Apr 1, 1982•523 citations
Starting from the exact result of Majumdar and Ghosh for the dimerized ground state of a Heisenberg linear-chain antiferromagnet with competing next-nearest-neighbor interaction J2{\mathit{J}}_{2}J2​=1/2J1{\mathit{J}}_{1}J1​, it is shown that this state is an exact eigenstate of the same Hamiltonian with couplings extended up to fourth-nearest neighbors, if the relations J1{\mathit{J}}_{1}J1​-2J2{\mathit{J}}_{2}J2​+J3{\mathit{J}}_{3}J3​=0 and J3{\mathit{J}}_{3}J3​=2J4{\mathit{J}}_{4}J4​ hold. Furthermore, it is proved to all orders in perturbation theory that this state is the ground state of this Hamiltonian in a finite interval of values of the parameter \ensuremath{\lambda}==1/2-j2{\mathit{j}}_{2}j2​ around \ensuremath{\lambda}=0. The actual extension of this interval is computed numerically for finite chains and is shown to be significant.
Starting from the exact result of Majumdar and Ghosh (1969) for the dimerized ground state of a Heisenberg S=1/2 antiferromagnet with competing next-nearest-neighbour interaction J2+1/2J1 on a linear chain, it is shown that this state is the ground state of the same kind of Hamiltonian, if couplings are extended up to 2n-nearest-neighbours, and the relation J1=2n, J2=2n-1, ..., J2n=1 holds. Furthermore, it is proved rigorously that this state is the ground state when n=2 and J1-2J2+J3=0, J3=2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0<or=J4<or=1/4J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising model is discussed.
We give an explicit proof that the completely dimerized (’Kekulè or nearest-neighbor resonant valence bond) state is the degenerate ground state of a linear chain S =1/2 Heisenberg antiferromagnet, with competing couplings extending up to 2n nearest neighbors, if the relation J1 = 2n, J2 = 2n − 1,...,J2n = 1, holds between the various exchange integrals. The previously known result for J2 =1/2 J1 is then just a special case corresponding to n=1. We study in detail n=2 (fourth-nearest neighbors) and show that this state is the ground state of the Hamiltonian when J1 − 2J2 + J3 = 0, J3 = 2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0 ≤ J4 ≤1/4 J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising models is made.
An antiferromagnetic spin S=1/2 model, the J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​-J3{\mathit{J}}_{3}J3​-J4{\mathit{J}}_{4}J4​-J5{\mathit{J}}_{5}J5​ model, is considered in two dimensions. J1{\mathit{J}}_{1}J1​,J2{\mathit{J}}_{2}J2​,J3{\mathit{J}}_{3}J3​,J4{\mathit{J}}_{4}J4​,J5{\mathit{J}}_{5}J5​ are the strengths of the nearest-neighbor, diagonal, next-nearest-neighbor, knight's-move-distance-away, and further-neighbor-diagonal interactions, respectively. It is known that when J1{\mathit{J}}_{1}J1​:J2{\mathit{J}}_{2}J2​:J3{\mathit{J}}_{3}J3​:J4{\mathit{J}}_{4}J4​:J5{\mathit{J}}_{5}J5​=1:1:1/2:1/2:1/4 the four columnar dimer states are the exact eigenstates. In a bosonic mean-field theory, we find that the columnar dimer state is the ground state. We compare the results obtained with those obtained in the same mean-field theory for the one-dimensional Majumdar-Ghosh (MG) model. For special values of the couplings, the dimer state is known to be the exact ground state of the MG model. The comparison of the mean-field results for both the models leads to the conjecture that the columnar dimer states are the exact ground states of the J1{\mathit{J}}_{1}J1​-J2{\mathit{J}}_{2}J2​-J3{\mathit{J}}_{3}J3​-J4{\mathit{J}}_{4}J4​-J5{\mathit{J}}_{5}J5​ model.
We study Heisenberg spin-1/2 and spin-1 chains with alternating ferromagnetic (J1FJ_1^FJ1F​) and antiferromagnetic (J1AJ_1^AJ1A​) nearest-neighbor interactions and ferromagnetic next-nearest-neighbor interaction (J2FJ_2^FJ2F​). In this model frustration is induced by the non-zero J2FJ_2^FJ2F​ and the effect of dimerization is realized by changing the strength of J1AJ_1^AJ1A​ for a fixed value of J1FJ_1^FJ1F​. The model with site spin sss behaves like a Haldane spin chain with site spin 2sss in the limit of vanishing J2FJ_2^FJ2F​ and large J1F/J1AJ_1^F/J_1^AJ1F​/J1A​. We show that the exact ground state of the model can be found along a line in the parameter space. The phase diagram in the space of J1A−J2FJ_1^A-J_2^FJ1A​−J2F​ is determined using numerical techniques complemented by some analytical calculations. A number of quantities, including the ground state spin, energy gap, structure factor, entanglement entropy, and zero temperature magnetization, are studied to understand the complete phase diagram. An interesting and application-wise important feature of this model is that it exhibits a macroscopic magnetization jump in the presence of a magnetic field; we study this using an effective Hamiltonian.
We perform a systematic DMRG investigation of the two natural spin-1 generalizations of the spin-1/2 Majumdar-Ghosh chain, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg chain, where J2{J}_{2}J2​ is a next-nearest-neighbor Heisenberg coupling, and the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, where J3{J}_{3}J3​ refers to a three-site interaction defined by ${J}_{3}\left[({\mathbf{S}}_{i\ensuremath{-}1}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i})({\mathbf{S}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{S}}_{i+1})+\mathrm{H}.\mathrm{c}.\right]$. Although both models are rigorously equivalent to the Majumdar-Ghosh chain for spin 1/2, their physics appears to be quite different for spin 1. Indeed, when all couplings are antiferromagnetic, the spin-1 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model undergoes an effective decoupling into two next-nearest-neighbor (NNN) Haldane chains upon increasing J2{J}_{2}J2​, while the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ chain undergoes a spontaneous dimerization similar to the spin-1/2 Majumdar-Ghosh chain upon increasing J3{J}_{3}J3​. By extending the phase diagram to all signs of the couplings, we show that both the dimerized and the NNN-Haldane phase are actually present in the ${J}_{1}\text{\ensuremath{-}}{J}_{3}$ model, the former one adjacent to the Haldane one, the latter to the ferromagneric one, with an Ising transition between them. By contrast, the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chain only has a NNN-Haldane phase between the Haldane phase and the ferromagnetic phase for positive J2{J}_{2}J2​. In both cases, our DMRG data are consistent with a continuous Kosterlitz-Thouless transition between the NNN-Haldane and the ferromagnetic phases.
Spontaneous dimerization is found in the S=12S=\frac{1}{2}S=21​ isotropic Heisenberg antiferromagnetic chain with competing nearest- and next-nearest-neighbor exchange, $\frac{{J}_{2}}{{J}_{1}}\ensuremath{\gtrsim}\frac{1}{6}$, and results from the same umklapp processes that lead to the N\'eel state when easy-axis exchange anisotropy is present. Spontaneous and externally induced dimerizations are contrasted.